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Properties

Label 2-627-11.3-c1-0-13
Degree $2$
Conductor $627$
Sign $0.942 + 0.333i$
Analytic cond. $5.00662$
Root an. cond. $2.23754$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.00 + 0.730i)2-s + (−0.309 − 0.951i)3-s + (−0.141 + 0.434i)4-s + (−2.10 − 1.53i)5-s + (1.00 + 0.730i)6-s + (−0.985 + 3.03i)7-s + (−0.943 − 2.90i)8-s + (−0.809 + 0.587i)9-s + 3.23·10-s + (−1.95 + 2.67i)11-s + 0.457·12-s + (3.54 − 2.57i)13-s + (−1.22 − 3.76i)14-s + (−0.804 + 2.47i)15-s + (2.32 + 1.69i)16-s + (−1.88 − 1.36i)17-s + ⋯
L(s)  = 1  + (−0.710 + 0.516i)2-s + (−0.178 − 0.549i)3-s + (−0.0706 + 0.217i)4-s + (−0.941 − 0.684i)5-s + (0.410 + 0.298i)6-s + (−0.372 + 1.14i)7-s + (−0.333 − 1.02i)8-s + (−0.269 + 0.195i)9-s + 1.02·10-s + (−0.589 + 0.807i)11-s + 0.131·12-s + (0.984 − 0.715i)13-s + (−0.327 − 1.00i)14-s + (−0.207 + 0.639i)15-s + (0.581 + 0.422i)16-s + (−0.456 − 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(627\)    =    \(3 \cdot 11 \cdot 19\)
Sign: $0.942 + 0.333i$
Analytic conductor: \(5.00662\)
Root analytic conductor: \(2.23754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{627} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 627,\ (\ :1/2),\ 0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.566209 - 0.0973185i\)
\(L(\frac12)\) \(\approx\) \(0.566209 - 0.0973185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (1.95 - 2.67i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (1.00 - 0.730i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (2.10 + 1.53i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.985 - 3.03i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.54 + 2.57i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.88 + 1.36i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 - 5.10T + 23T^{2} \)
29 \( 1 + (-1.44 + 4.45i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.62 + 3.36i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.576 + 1.77i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.14 + 9.66i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.884T + 43T^{2} \)
47 \( 1 + (-2.22 - 6.85i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-11.2 + 8.18i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.155 - 0.478i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.91 - 4.30i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 5.78T + 67T^{2} \)
71 \( 1 + (7.99 + 5.80i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.27 + 3.91i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.44 - 3.22i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.62 - 4.81i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (7.21 - 5.24i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.42172208596170741044528499270, −9.296482634445965078069452593429, −8.600856722208146534904471983378, −8.043465322772991427476102998903, −7.25522660106681961673920544203, −6.25845535374677174209695323254, −5.21073386076022846817247399108, −3.95011100457924799567033845700, −2.62130420044246297669000773651, −0.56948418796052663398968181259, 0.942083715350871473284175062471, 2.98414534103966590761688922743, 3.84106085847018691085615057845, 4.96447163439017372885676375069, 6.30180971906075091032679297005, 7.14726939146480197377897595603, 8.340685915908666366393508595307, 8.924327746010787252974360009967, 10.11901719221860476301540528408, 10.63552083187903054384127531971

Graph of the $Z$-function along the critical line